%Introductory text: explain the problem around uncertainty. General things. Announce what will come.
%Remember: clearly state that everything before this point concerned the crisp case.


%Possibility theory: general things. DO NOT FORGET THE INTERPRETATION PART.


%Possibilistic variables


%Ill-known constraints. Make certain it is clear that entire sets can be evaluated, but we only use intervals...


%Fuzzy numbers and intervals. Give an introduction to make the transition smooth



\subsection{\label{subsec:possibility-theory}Possibility Theory}
Possibility theory, like probability theory, deals with uncertainty about the outcome of an experiment. In probability theory, this uncertainty is caused by the \emph{variability} in the outcomes, while in possibility theory, the uncertainty is caused by \emph{incomplete knowledge} about the experiment. The quantification of confidence in a theory of uncertainty is achieved using a confidence measure\cite{Pons2011}. In probability theory this is a measure of chance, in possibility theory, possibility and necessity measures are used.

\begin{definition}
Consider a set of outcomes $\Omega$. Let $\wp(\Omega)$ denote the powerset of $\Omega$ and let $A$ and $B$ be elements of $\wp(\Omega)$. A \emph{confidence measure on $\Omega$} is defined by a function
	\begin{align}
	g : \wp(\Omega) & \rightarrow \left[0,1\right]
	\end{align}
that satisfies
	\begin{align}
	g(\emptyset) &= 0 \\
	g(\Omega) &= 1 	\label{NormalizationProperty} \\
	A \subseteq B &\Rightarrow g(A) \leq g(B) \label{MonotonicityProperty}
	\end{align}
\end{definition}

Both possibility measures and necessity measures are special cases of confidence measures.

\begin{definition}
Consider a confidence measure $\Pi$ on a set of outcomes $\Omega$. Let $J$ be a countable index set and let $\{ A_{j} | j \in J \wedge A_{j} \subseteq \Omega \}$ be a family of elements of $\wp(\Omega)$. $\Pi$ is now a \emph{possibility measure on $\Omega$} if it satisfies:
	\begin{align}
	\Pi\left(\bigcup_{j \in J} A_{j} \right) = \sup_{j \in J} \Pi(A_{j})
	\end{align}
\end{definition}

In this work, the interpretation is as follows. The possibility of an event expresses how plausible the occurrence of the event seems to an observer of the experiment, given the (partial) knowledge of the observer about the experiment.

Information on the possibility of distinct elements of the universe of discourse $\Omega$ can now be given by a \emph{possibility distribution} $\pi$ on $\Omega$, defined by:

\begin{definition}
Consider a possibility measure $\Pi$ on $\Omega$. A \emph{possibility distribution} $\pi$ on $\Omega$ underlying the possibility measure $\Pi$ is then a function defined by:
	\begin{align}
	\pi : \Omega \rightarrow \left[0, 1\right] : \pi(u) = \Pi(\{u\})
	\end{align}
\end{definition}

\begin{definition}
Consider a confidence measure $N$ on a set of outcomes $\Omega$. Let $J$ be a countable index set and let $\{ A_{j} | j \in J \wedge A_{j} \subseteq \Omega \}$ be a family of elements of $\wp(\Omega)$. $N$ is now a \emph{necessity measure} on $\Omega$ if it satisfies:
	\begin{align}
	N\left(\bigcap_{j \in J} A_{j} \right) = \inf_{j \in J} N(A_{j})
	\end{align}
\end{definition}

In this work, the interpretation is as follows. The necessity of an event expresses how necessary the occurrence of the event seems to an observer of the experiment, given the (partial) knowledge of the observer about the experiment.

Possibility and necessity measures are dual in the sense that:

\begin{align}
\forall A \subseteq \Omega : N(A) = 1 - \Pi(\bar{A})
\end{align}

Regarding interpretation, the above can be seen as: the degree to which an event is necessary is the degree to which every other possible event is not plausible.

\subsection{\label{subsec:possibilistic-variables}Possibilistic Variables}
Possibilistic variables rely on possibility theory \cite{Dubois1988a}. A \emph{possibilistic variable} is defined as follows \cite{Pons2011}.

\begin{definition}
\label{def;possibilistic-variable}
A possibilistic variable $X$ over a universe $U$ is defined as a variable taking exactly one value in $U$, but for which this value is (partially) unknown. Its possibility distribution $\pi_X$ on $U$ models the available knowledge about the value that $X$ takes: for each $u\in U$, $\pi_X(u)$ represents the possibility that $X$ takes the value $u$. In this work, this possibility is interpreted as a measure of how plausible it is that $X$ takes the value $u$, given (partial) knowledge about the value $X$ takes.
\end{definition}

The exact value a possibilistic variable takes, which is (partially) unknown, is called an \emph{ill-known value} in this work \cite{Dubois1988a}.

When a possibilistic variable is defined on the powerset $\Pow(R)$ of some universe $R$, the unique value the variable takes will be a crisp set and its possibility distribution on the powerset $\Pow(R)$ will describe the possibility of each crisp subset of $R$ to be the value the variable takes. This exact value (a crisp set) the variable takes, is now called an \emph{ill-known set} \cite{Dubois1988a}.

It is important to understand the difference between the following two concepts:
\begin{itemize}
\item
A \emph{possibilistic variable} $X$ is bounded to take only one value , but this value is not known due to incomplete knowledge. 
\item
An \emph{ill-known set}~\cite{Dubois1988a}: a possibilistic variable defined over the universe $\Pow(U)$.
\end{itemize}

Note that while a possibilistic variable refers to one (partially) unknown value, an ill-known set is a crisp set but, for some reason, (partially) unknown.

A specific application of possibilistic variables is obtained when the universe under consideration is the set of Boolean values $\mathbb{B}$ = $\{T,F\}$. Indeed, any Boolean proposition $p$ takes just one value in $\mathbb{B}$. If the knowledge about which value this proposition $p$ will take is given by a possibility distribution $\pi_p$, the proposition can be seen as a possibilistic variable. As the interest lies with the case where the proposition holds, the possibility and necessity that $p$ = $T$ (the proposition holds) demand most attention. This possibility and necessity is noted here as:
\begin{align}
\label{propholdsposs}
\text{Possibility that $p$ = $T$ (p holds):} &\\
\nonumber
Pos(p) = \pi_p(T)  \\
\label{propholdsnecc}
\text{Necessity that $p$ = $T$ (p holds):} & \\
\nonumber
Nec(p) = 1-\pi_p(F) 
\end{align}

Here, equation \eqref{propholdsposs} denotes the possibility that $p$ = $T$ and the proposition holds, equation \eqref{propholdsnecc} denotes the necessity that $p$ = $T$ and the proposition holds.

This work will deal with ill-known intervals. These are ill-known sets, defined and represented via a start and end point, which will be ill-known values. The elements of the set are the values between the starting and ending points. A closed ill-known interval with starting point defined by possibilistic variable $X$ and ending point by possibilistic variable $Y$ is noted here $\left[X, Y\right]$. The correspondences and transitions between the representations of ill-known sets, between the representations of ill-known intervals and between the representations of an ill-known set and an ill-known interval are part of the authors current research.

\subsection{\label{subsec:fuzzy-numbers}Fuzzy Numbers and Fuzzy Intervals}
Among others, Dubois and Prade~\cite{Dubois1983} use fuzzy sets \cite{Zadeh1965} to define a \emph{fuzzy interval}:
\begin{definition}
A fuzzy interval is a fuzzy set $M$, defined by a membership function $\mu_{M}$, on the set of real numbers $\mathbb{R}$ such that:
\begin{eqnarray}
\mu_{M} :  \mathbb{R} \rightarrow \left[0,1\right] \nonumber \\ 
\forall (u,v)\in\mathbb{R}^2: \forall w \in [u,v]:\\
\mu_M(w) \geq\min(\mu_M(u),\mu_M(v))  \\
\exists m \in \mathbb{R} :  \mu_M(m)=1 
\end{eqnarray}
\end{definition}

\vspace*{13pt}
\begin{center}
{
\includegraphics[scale=0.25]{./graphs/triangular.pdf}

}
\end{center}
%\centerline{ \psfig{file=./graphs/Y-time-point.eps}}
\vspace*{10pt}
\fcaption{\label{fig:triangular-dist}Example of fuzzy number.}
%  \label{fig:fuzzy-validity-period}
\vspace*{13pt}

If this modal value $m$ is unique, then $M$ is referred to as a \emph{fuzzy number}. 
%In other words, if the core of a fuzzy interval is a singleton, it is referred to as a fuzzy number (Figure \ref{fig:triangular-dist}).

A simple form of the membership function of a fuzzy interval is a trapezoidal function (Figure \ref{fig:trapezoidal-dist}). It can be shown that such a membership function $\mu_T$ for a fuzzy interval $T$ is convex and normalized. Four real values, denoted $\alpha$, $\beta$, $\gamma$ and $\delta$ suffice to represent a trapezoidal membership function of a fuzzy interval. In this work, a fuzzy interval defined as such will be noted as $\left[\alpha, \beta, \gamma, \delta\right]$. The corresponding membership function definition for this $\mu_T$ is then given by:

\begin{align}
\mu_T : & \quad \mathbb{R} \rightarrow \left[0,1\right] \\
 : & \quad x \rightarrow
\begin{cases}
1 & \mbox{ if } x \in [\beta,\gamma] \\
0 & \mbox{ if } x > \delta \vee x < \alpha \\
\frac{x-\alpha}{\beta - \alpha} & \mbox{ if } x \in [\alpha,\beta[ \\
\frac{\delta -x}{\delta - \gamma} & \mbox{ if } x \in ]\gamma,\delta] \\
\end{cases}
\end{align}


\vspace*{13pt}
\begin{center}
{
\includegraphics[scale=0.25]{./graphs/trapezoidalDistribution.pdf}

}
\end{center}
%\centerline{ \psfig{file=./graphs/Y-time-point.eps}}
\vspace*{10pt}
\fcaption{\label{fig:trapezoidal-dist}Example of fuzzy interval.}
%  \label{fig:fuzzy-validity-period}
\vspace*{13pt}
%\def\JPicScale{0.5}
%\begin{figure}[h!]
%  \centering
%  \input{./graphs/trapezoidal.tex}
%  \caption{Trapezoidal membership function}
%  \label{fig:trapezoidal}
%\end{figure}

The most convenient form of the membership function of a fuzzy number is a triangular form. It can be shown that such a membership function $\mu_M$ for a fuzzy number $M$ is convex and normalized. Three real values, denoted by $a$, $b$ and $D$, suffice to represent a triangular membership function of a fuzzy number and in this work, a fuzzy number defined as such will be noted as $\left[D, a, b \right]$. Here:
\begin{itemize}
\item
$D$ denotes the single value in the core of $M$
\item
$D-a$ is then $\inf \{u \in \mathbb{R} : \mu_{M}(u) > 0\}$
\item
$D+b$ is then $\sup \{u \in \mathbb{R} : \mu_{M}(u) > 0\}$
\end{itemize}

The membership function of such a fuzzy number is then given by:

\vspace{-10pt}

\begin{align}
\mu_M :   \mathbb{R} \rightarrow \left[0,1\right] :  x \rightarrow \nonumber\\
\begin{cases}
\nonumber
0 & \mbox{ if } \left(x < D-a\right) \vee \left(x > D+b\right) \\
\frac{\left[x-\left(D-a\right)\right]}{a} & \mbox{ if } \left(x \geq D-a\right) \wedge (x \leq D)  \\
-\frac{\left[x-\left(D+b\right)\right]}{b} & \mbox{ if } \left(x \geq D\right) \wedge (x \leq D+b)
\end{cases}
\end{align}

%\begin{figure}[h!]
%  \centering
%  \input{./graphs/triangular.tex}
%  \caption{Triangular membership function.}
%  \label{fig:triangular}
%\end{figure}



\subsection{\label{subsec:interval-evaluation-by-ill-known-constraints}Interval Evaluation by Ill-known Constraints}
The problem of interval evaluation is more generally explained in \cite{Pons2011}: the need exists to know if all points in a crisp interval $I$ reside between the boundaries of an ill-known interval $\left[ X , Y \right]$. In \cite{Pons2011}, the notion of an \emph{ill-known constraint} is introduced:

\begin{definition}
Given a universe $U$, an ill-known constraint $C$ on a set $A \subseteq U$ is specified by means of a binary relation $R \subseteq U^{2}$ and a fixed, ill-known value denoted by its possibilistic variable $V$ over $U$, i.e.:
\begin{align}
\label{eq:ill-known-constraint}
C \triangleq (R,V)
\end{align}
Set $A$ now satisfies the constraint if and only if:
\begin{align}
\forall a \in A : (a,V) \in R
\end{align}
\end{definition}

An example of an ill-known constraint is $C_{ex} \triangleq (<, X)$. Some set $A$ then satisfies $C_{ex}$ if $\forall a \in A : a < X$, given possibilistic variable $X$.

The satisfaction of a constraint $C \triangleq (R,V)$ by a set $A$ is basically still a Boolean matter, but due to the uncertainty about the ill-known value $V$, it can be uncertain whether $C$ is satisfied by $A$ or not \cite{Pons2011}. In fact, this satisfaction now behaves as a proposition. Based on the possibility distribution $\pi_{V}$ of $V$, the possibility and necessity that $A$ satisfies $C$ can be found. This proposition can thus be seen as a possibilistic variable on $\mathbb{B}$. The required possibility and necessity are:

\vspace{-10pt}

\begin{align}
\Pos(A\text{ satisfies }C) & =\\
\nonumber
\min_{a \in A}\left(\sup_{(a,w) \in R}\pi_{V}(w)\right) \label{ill-known-pos}\\
\Nec(A\text{ satisfies }C) & =\\
\nonumber
\min_{a \in A}\left(\inf_{(a,w) \notin R} 1-\pi_{V}(w)\right) \label{ill-known-nec}
\end{align}

So far, we have shown how it can be verified if a set satisfies or not an ill-known constraint.

\begin{example}

\vspace*{13pt}
\begin{center}
{
\includegraphics[scale=1]{./graphs/example-ill-known.pdf}
}
\end{center}
%\centerline{ \psfig{file=./graphs/Y-time-point.eps}}
\vspace*{10pt}
\fcaption{\label{fig:example-ill-known}Example of the evaluation of the ill-known constraint $C \triangleq (\leq, X)$. Possibility and necessity measures are shown in grey the upper and lower graphics respectively. }
%  \label{fig:fuzzy-validity-period}
\vspace*{13pt}


 Consider an ill-known value given by $X = \left[5, 3, 2 \right]$. $C \triangleq (\leq, X)$ is the ill-known constraint. The set $A$ is $A \subseteq \R^+$. Then, the evaluation of the possibility and the neccesity are obtained from \eqref{ill-known-pos} and \eqref{ill-known-nec} respectively. (See Figure \ref{fig:example-ill-known}).
 
\begin{align}
\Pos(A\text{ satisfies }C) & =\\
\nonumber
\min_{a \in A}\left(\sup_{a \leq R}\pi_{X}(w)\right)\\
\Nec(A\text{ satisfies }C) & =\\
\nonumber
\min_{a \in A}\left(\inf_{a > w} 1-\pi_{X}(w)\right) 
\end{align}




\end{example}





 However, from the examples at the beginning of this section, it is observed that Boolean combinations of constraints are required. For example, the problem of interval evaluation as explained earlier requires that all elements of an interval $[a,b]$ are larger than a value $X$ and at the same time smaller than a value $Y$, which implies that a conjunctive Boolean combination of both constraints must be satisfied. To allow Boolean combinations of constraints, the following definitions are introduced.
\begin{definition}
\label{def:set-evaluation}
Consider a universe $U$, an $n$-ary vector $\mathbf{C}$ of constraints and a Boolean function $\bool:\mathbb{B}^{n}\rightarrow\mathbb{B}$. An evaluation function is defined by:
\begin{equation}
\lambda:\Pow(U)\rightarrow\mathbb{B}:A\mapsto\bool\Big(C_{1}(A),...,C_{n}(A)\Big).
\end{equation}
\end{definition}
Definition \ref{def:set-evaluation} presents the definition of an evaluation function that evaluates a Boolean combination of some basic constraints. Informally, it states that a set $A$ passes the evaluation made by $\lambda$ if the Boolean combination of some propositions equals $T$. This crisp definition can be generalized to the case of ill-known constraints.
\begin{definition}
\label{def:ill-known-sets}
Consider a universe $U$, an $n$-ary vector $\mathbf{C}$ of ill-known constraints and a Boolean function $\bool:\mathbb{B}^{n}\rightarrow\mathbb{B}$. The uncertainty about the evaluation of a set $A$ by an evaluation function $\lambda$ is then given by:
\begin{equation}
\forall A\in\Pow(U):\pi_{\lambda(A)}=\widetilde{\bool}\Big(\pi_{C_1(A)},...,\pi_{C_n(A)}\Big)\\
\end{equation}
Hereby, $\widetilde{\bool}$ is the possibilistic extension of $\bool$.
\end{definition}
It is well known that any Boolean function $\bool$ can be cast to a canonical form~\cite{McCluskey1965}, requiring only the logical conjunction $\wedge$, logical disjunction $\vee$ and logical negation. Therefore, only the case of Boolean conjunction, Boolean disjunction and Boolean negation will be treated within the scope of this paper. By applying the possibilistic extensions of $\wedge$, $\vee$ and $\neg$, concrete equations are obtained for the calculations of uncertainty about the evaluation of a set by means of an evaluation function $\lambda$. In the case of conjunction (i.e., $\bool=\wedge$), the inference of uncertainty about the evaluation of a set reduces to:
\begin{eqnarray}
\label{eq:conjunctive1}
\forall A\in\Pow(U):\Pos(\lambda(A))&=\\
\nonumber
\min_{i=1}^n\Pos\left(C_i(A)\right)\\
\label{eq:conjunctive2}
\forall A\in\Pow(U):\Nec(\lambda(A))&=\\
\nonumber
\min_{i=1}^n\Nec\left(C_i(A)\right).
\end{eqnarray}
In the case of disjunction (i.e. $\bool=\vee$), the inference of uncertainty about the evaluation of a set reduces to:
\begin{eqnarray}
\label{eq:disjunctive}
\forall A\in\Pow(U):\Pos(\lambda(A))&=\\
\nonumber
\max_{i=1}^n\Pos\left(C_i(A)\right)\\
\forall A\in\Pow(U):\Nec(\lambda(A))&=\\
\nonumber
\max_{i=1}^n\Nec\left(C_i(A)\right).
\end{eqnarray}
Note that by using the functions $\min$ and $\max$ here, there is an implicit assumption that the possibilistic variables $\pi_{C_i}$ are mutual $\min$-dependent in the sense of De Cooman (i.e. non-interactive). For an extensive reading on (in)dependency of possibilistic variables, the reader is referred to~\cite{GertDeCooman1997b},\cite{GertDeCooman1997a},\cite{GertDeCooman1997}. In case of $\neg$, we get:
\begin{eqnarray}
\label{eq:negation}
\forall A\in\Pow(U):\Pos(\neg\lambda(A))&=\\
\nonumber
1-\Nec(\lambda(A))\\
\forall A\in\Pow(U):\Nec(\neg\lambda(A))&=\\
\nonumber
1-\Pos(\lambda(A)).
\end{eqnarray}

\begin{example}
Consider that we want to check if crisp interval $I = \left[j, k\right]$ is included in $\left[X, Y\right]$, 2 ill-known constraints are constructed:

%We assume that $X$ specifies the lower bound and $Y$ the upper bound for a given interval, we want to known whether all points in the interval are larger than or equal to $X$ and smaller than or equal to $Y$. Therefore, we consider two ill-known constraints:

\vspace{-10pt}

\begin{eqnarray}
C_1 & \triangleq\left(\geq,X\right)\\
C_2 & \triangleq\left(\leq,Y\right)
\end{eqnarray}

To calculate the possibility and necessity concerning a conjunction of constraints, the $\min$ operator can be used. The possibility and necessity of $I$ being included in $\left[X, Y\right]$ are now: %satisfying both constraints is then:

\vspace{-10pt}

\begin{align}
\label{eq:interval-pos}
\Pos(I\text{ satisfies }C_1\ and\ C_2) & =\\
\nonumber
\min_{a \in I}\left(\sup_{a \geq w}\pi_{X}(w),\sup_{a \leq v}\pi_{Y}(v)\right)\\
\label{eq:interval-nec}
\Nec(I\text{ satisfies }C_1\ and\ C_2) & =\\
\nonumber
\min_{a \in I}\left(\inf_{a < w} 1-\pi_{X}(w),\inf_{a > v} 1-\pi_{Y}(v)\right).
\end{align}
\end{example}

There is a special boolean combination of constraints that is of particular interest:
\begin{definition}
\emph{Convex combination of ill-known constraints}. Consider the $C_1$ and $C_2$ two ill-known constraints. A convex combination of the constraints is given by:
\begin{align}
\label{eq:convex-combination}
CC \triangleq \left \lbrace C_1 \wedge C_2 \right \rbrace
\end{align}
In a more general way, it is possible to define the convex combination of an n-ary vector  $\mathbf{C}$ of constraints:
\begin{align}
\label{eq:nary-convex-combination}
CC \triangleq \left \lbrace C_1 \wedge \ldots \wedge C_n \right \rbrace
\end{align}
\end{definition}

\begin{theorem}
\label{th:convex-combination-ill-known-constraints}
Consider the convex combination $CC$ of any n-ary vector $\mathbf{C} = \left \lbrace C_{1_{Z_1}}, \ldots, C_{n_{Z_n}} \right \rbrace$ of constraints over the ill-known variables $Z_1, \ldots, Z_n$. Then, if $\pi_{C_1(Z_1)}, \ldots, \pi_{C_n(Z_n)}$ are convex, then  $\pi_{CC}$ is also convex.
\end{theorem}

\begin{proof}
Let $CC \triangleq \left \lbrace C_{1_{Z_1}}, \ldots, C_{n_{Z_n}}  \right \rbrace$. Then:
\begin{align}
\label{eq:proof-convex1}
\pi_{CC} \left( \lambda x_1 + \left( 1 - \lambda \right)x_2 \right) = \\
\min  \big( \pi_{C_1(Z_1)}\left( \lambda x_1 + \left( 1 - \lambda \right)x_2 \right), \ldots,\\
   \pi_{C_n(Z_n)}\left( \lambda x_1 + \left( 1 - \lambda \right)x_2 \right)  \big)
\end{align}
Since $\pi_{C_1(Z_1)}, \ldots, \pi_{C_n(Z_n)}$ are convex:
\begin{align}
\label{eq:proof-convex2}
\pi_{C_1(Z_1)} \left(\lambda x_1 + \left( 1 - \lambda \right)x_2  \right) \geqq \\
\nonumber
\min \big(\pi_{C_1(Z_1)} \left( x_1 \right),\pi_{C_1(Z_1)} \left( x_2 \right)  \big)\\
\nonumber
\vdots \\
\nonumber
\pi_{C_n(Z_n)} \left(\lambda x_1 + \left( 1 - \lambda \right)x_2  \right) \geqq \\
\nonumber
\min \big(\pi_{C_n(Z_n)} \left( x_1 \right),\pi_{C_n(Z_n)} \left( x_2 \right) \big) 
\end{align}
Then, by using equation \eqref{eq:proof-convex1}:
\begin{align}
\label{eq:proof-convex3}
\pi_{CC} \left( \lambda x_1 + \left( 1 - \lambda \right)x_2 \right) \geqq  \\
\nonumber
\min \big( \min \left(\pi_{C_1(Z_1)} \left( x_1 \right),\pi_{C_1(Z_1)} \left( x_2 \right) \right),\\
\nonumber
\ldots, \min \left(\pi_{C_n(Z_n)} \left( x_1 \right),\pi_{C_n(Z_n)} \left( x_2 \right) \right) \big)
\end{align}
Which is equivalent to the following:
\begin{align}
\label{eq:proof-convex4}
\pi_{CC} \left( \lambda x_1 + \left( 1 - \lambda \right)x_2 \right) \geqq  \\
\nonumber
\min \big( \min \left(\pi_{C_1(Z_1)} \left( x_1 \right), \ldots, \pi_{C_n(Z_n)} \left( x_1 \right) \right),\\
\nonumber
\ldots, \min \left(\pi_{C_1(Z_1)} \left( x_2 \right), \ldots, \pi_{C_n(Z_n)} \left( x_2 \right) \right) \big)
\end{align}
Finally we obtain:
\begin{align}
\label{eq:proof-convex5}
\pi_{CC} \left( \lambda x_1 + \left( 1 - \lambda \right)x_2 \right) \geqq  \\
\nonumber
\min \big(\pi_{CC} \left( x_1 \right), \pi_{CC} \left( x_2 \right) \big)
\end{align}
\end{proof}

Sometimes, an ill-known value might be specified by a convex combination of ill-known constraints. This allow to define ill-known values by means of relationships with respect to other ill-known points.

%\begin{definition}
%\emph{Ill-known point defined by ill-known constraints}
%Consider a universe $U$, an n-ary vector $\mathbf{C}$ of ill-known constraints and a Boolean function $\bool:\mathbb{B}^{n}\rightarrow\mathbb{B}$. An ill-known value $X$ is defined by:
%\begin{align}
%\label{eq:ill-known-value-def-by-const}
%X \in \Pow(U):\pi_{X}=\widetilde{\bool}\Big(\pi_{C_1(Z_1)},...,\pi_{C_n(Z_n)}\Big)\\
%\end{align} 
%\end{definition}

\begin{definition}
\label{def:convex-combination-ill-known-constraints}
Consider a universe $U$, $X$ an ill-known value and $CC \triangleq \left \lbrace C_{1_{Z_1}}, \ldots, C_{n_{Z_n}}  \right \rbrace$ a convex combination of ill-known constraints over the variables $Z_1, \ldots, Z_n$. The value $X$ is defined by means of the convex combination:
\begin{align}
\label{eq:ill-known-value-by-convex-constraints}
X \triangleq CC 
\end{align}

The uncertainty about the evaluation of an ill-known value $X$ is given by:
\begin{align}
\label{eq:ill-known-value-def-by-const-unc}
X \in \Pow(U):\pi_{X}=\pi_{CC}
\end{align} 
Note that $\pi_{X}$ is convex since $\pi_{CC}$ is convex as demonstrated in theorem \ref{th:convex-combination-ill-known-constraints}.
\end{definition}


\begin{example}
Consider the ill-known values $X = \left[12, 2, 2\right]$ and $Y = \left[18, 2, 1 \right]$. The ill-known value $Z$ is defined using the convex combination $CC$ of constraints $C_1$ and $C_2$:
\begin{align}
\nonumber
CC \triangleq \left \lbrace C_1\left(>,X\right) \wedge C_2(\leq,Y) \right \rbrace \\
\nonumber
Z \triangleq CC
\end{align}
$Z$ is a fuzzy interval defined by a trapezoidal shape given by $\left[12,\ 14,\ 18,\ 19 \right]$.
Figure \ref{fig:example-ill-known-by-const} illustrates the relations among the variables $X$, $Y$ and $Z$.
\end{example}

\vspace*{13pt}
\begin{center}
{
\includegraphics[scale=1]{./graphs/ill-known-by-constraints.pdf}

}
\end{center}
%\centerline{ \psfig{file=./graphs/Y-time-point.eps}}
\vspace*{10pt}
\fcaption{\label{fig:example-ill-known-by-const}Ill-known values $X$ and $Y$. The grey area represents the ill-known value $Z$ defined by the convex combination of the two ill-known constraints $C_1$ and $C_2$.}
\vspace*{13pt}

%\paragraph{Example} Consider the ill-known values $X = \left[5, 2, 8\right]$ and $Y = \left[9, 7, 10 \right]$. The knowledge about the evaluation of the interval $\left[a, b \right]$  is given by the expressions \eqref{eq:interval-pos},\eqref{eq:interval-nec}.  Figure~\ref{fig:3d-possibility} shows a 3D plot of the possibility that an interval $[a,b]$ passes the evaluations specified by the ill-known constraints. Note the triangular form for the resulting possibility distribution since the condition $a \leq b$ holds.
%
%\begin{figure}[h!]
%\centering
%\includegraphics[scale=0.4]{graphs/3D_possibility.eps}
%\caption{Possibility of evaluation for the interval $[a,b]$.}
%\label{fig:3d-possibility}
%\end{figure}
%The necessity plot is obtained in a similar way and is shown in Figure~\ref{fig:3d-necessity}. Notice that the necessity measure is not normalized because the supports of $X$ and $Y$ overlap.
%\begin{figure}[h!]
%\centering
%\includegraphics[scale=0.4]{graphs/3D_necessity.eps}
%\caption{Necessity of evaluation for the interval $[a,b]$.}
%\label{fig:3d-necessity}
%\end{figure}




